Modes of deformation

At temperatures of interest to the engineer, polycrystalline ice is an elastic-visco-plastic solid. Each grain is anisotropic with respect to its elastic properties. It has now been amply demonstrated that it is also anisotropic with respect to its viscous and plastic behaviour; the shearing resistance of non-basal planes being a factor of more than ten greater than that of the basal planes (Reference Higashi, Higashi, Mae and FukudaHigashi and others, 1968).

When a grain in polycrystalline ice is forced to undergo a constant-volume change in shape and the stress is not parallel or perpendicular to its axis of crystallographic symmetry, it does so primarily by slip on basal planes and rotation about low-angle boundaries that form perpendicular to them (Reference GoldGold, 1972[a]). Accommodation that must take place between grains during ductile deformation does so primarily in the grain-boundary regions. Here, evidence is found of grain-boundary migration, polygonization and void formation. With continued deformation this region of major distortion grows until it consumes the adjacent grains. If the stress is sufficiently high, this process is accompanied by the formation of internal cracks. It is clear that visco-plastic deformation of ice is associated with structural changes that can have a significant effect on its properties.

Transmission electron microscopy has shown that slip on the basal plane is associated with the stress-induced motion of dislocations (Reference Fukuda and HigashiFukuda and Higashi, 1973). Grain boundaries are potential barriers for dislocations, and can be expected to cause the formation of pile-ups. The stress near the head of a pile-up is directly proportional to the effective shear stress acting on it and to (l/r)½ where l is the length of the pile-up and r is the distance from the head to the point where the stress is being determined. It can be appreciated that during ductile deformation, the stress concentrations at such sites can become quite large and be responsible for the initiation of other modes of deformation, such as grain-boundary migration, climb of dislocations out of their slip planes and the formation of cracks. If the rate of change in shape of a grain is controlled by processes occurring primarily in the grain-boundary regions (e.g. climb or slip on non-basal planes) the effective stress acting on the pile-ups will approach the resolved shear stress on the basal plane.

The contribution to the strain-rate by the movement of dislocations is given by:

(1)

where is the strain rate, s^-1, ρ _m the density of mobile dislocations, m^-2, a function of stress and strain, b the Burgers vector of the mobile dislocations = 4.5 × 10^-10 m, the average velocity of the dislocations, and α a geometric factor. Observations of the yield behaviour of ice single crystals subject to constantrate of strain have shown that it is in agreement with the model presented by Reference JohnstonJohnston (1962) (Reference Higashi and OuraHigashi, 1967; Reference Jones and GlenJones and Glen, 1969). Yield is attributed in this model to an increase in the dislocation density with deformation and the stress dependence of the average dislocation velocity. The resistance to deformation of polycrystalline ice is appreciably greater than that for slip on the basal planes of single crystals, and yield for it appears to be associated with the initiation and development of other modes of deformation, particularly internal crack formation for stress greater than 1 MN m^-2 (Reference GoldGold, 1972[a]; Reference Carter and MichelCarter and Michel, 1971).

The information published on the stress dependence of the velocity of dislocations in ice is presented in Figure 1. These observations indicate that this dependence is given, approximately, by:

Fig. 1.

Stress dependence of the dislocation velocity oil the basal plane.

(2)

where τ is the shear stress acting on the dislocation. Equations (1) and (2) show that the greater the imposed rate of strain the larger must be the stress acting on mobile dislocations and, therefore, the greater the probability of crack formation and yield or fracture because of this activity.

If it is assumed that a typical grain dimension in the basal plane is about 2 × 10^-3 m and the stress dependence of the dislocation velocity is given by Equation (2), then it would take about 300 s for a dislocation to travel from the centre of the grain to a boundary under a shear stress of 1 MN m^-2. It would not be expected, therefore, that many dislocations would be forced out of their glide plane during the first 100 s of deformation for applied stress less than 2 MN m^-2. Reference Traetteberg, Traetteberg, Gold, Frederking and FrankensteinTraetteberg and others (1975) found that for this condition ice is essentially elastic in the sense that when the load is applied for less than 100 s, practically all the strain is recovered after the removal of the stress.

Polycrystalline ice can be expected to fail in a brittle manner when the strain-rate exceeds about 10^-3 s^-l. The unconfined compressive strength at this rate of strain is about 10 MN m^-2. A typical value of Young's modulus for this rate of strain is 7 × 10³ MN m^-2. Assuming that the shear stress on a dislocation increases linearly from 0 to 5 MN m^-2 during loading, the distance travelled by a dislocation during the time to failure, according to Equation (2), is about 10^-5 m and appreciably less than typical grain dimensions in the basal plane.

The foregoing discussion indicates the importance of an appreciation of the elastic, viscous and plastic properties of ice for engineering problems. Much of the behaviour of interest to engineers occurs over a range of strain-rate for which, at the low end, polycrystalline ice is quite ductile, particularly at temperatures greater than –20°C, and at the high end it is brittle. It is clear that the time, stress and temperature dependence of viscous processes, both on active slip planes and in grain-boundary regions and the effect of these processes on the deformation behaviour (e.g. yield, crack formation), play a controlling role in this transformation in behaviour.

Elastic moduli

The foregoing discussion, as well as laboratory measurements, indicates that ice can be assumed to respond elastically to stress when the period of application of the load is less than 100 s for stress less than 1 MN m^-2 (Reference Traetteberg, Traetteberg, Gold, Frederking and FrankensteinTraetteberg and others, 1975) or if loaded to failure within about 2 s. Most bearing-capacity problems for moving loads, and many ice-structure interaction problems, including ice breaking, satisfy these conditions. Knowledge concerning the elasticity of polycrystalline ice is not, however, very well developed. It is known that the elastic moduli depend upon several factors, including temperature, density, ice type, purity, time or frequency of stress, and direction of application of the load. There is evidence that they may depend also upon grain size and stress history. Figure 2 presents information that has been published on the frequency dependence of Young's modulus for polycrystalline ice at –l0°C. This information divides naturally into three regions: the frequency range below about 10Hz, which is the range presently covered by hydraulic and mechanical testing machines; the range from about 10² to 10⁴ Hz associated with the natural period of vibration of ice rods and propagation of seismic-type disturbances in ice covers or large ice bodies; and the range above about 105 Hz associated with measurements using ultrasonic pulse techniques. A significant gap exists between about 10 to 10² Hz, a very difficult range in which to make measurements.

Fig 2.

Frequency dependence of Young's modulus for polycrystalline ice at - 10°C.

It can also be seen from Figure 2 that Young's modulus increases with frequency at the low frequency range, tending to the values measured in the range 10² to 10⁴ Hz. Observations on the energy loss during cyclical stressing as well as a comparison of polycrystalline and single-crystal values of the elastic moduli, indicate that the frequency dependence in this region is due to relative movement between grains in the grain-boundary region (Reference Yamaji and KuroiwaYamaji and Kuroiwa, 1956; Reference GoldGold, 1958; Reference NakayaNakaya, 1959; Reference KuroiwaKuroiwa, 1964). Kuroiwa has shown that this effect is dependent on the grain-boundary area and the impurity content of the ice.

Both Yamaji and Kuroiwa, and Nakaya, observed a decrease in Young's modulus with increasing frequency in the frequency range of about 200 to 600 Hz. This is somewhat unusual, as the modulus would normally be expected to either remain constant or increase with frequency. Nakaya did report that the modulus of his samples increased after a rest period of about 12 h. This is shown in Figure 2 by the arrow between the two points plotted at about 575 Hz. The results of Yamaji and Kuroiwa are sufficiently consistent, however, to be considered as strong evidence that the decrease is real Nakaya's observations on the change in Young's modulus with time, and a change observed between the first and subsequent sets of measurements for a columnar-grained specimen in the frequency range below 10 Hz (see Fig. 2) do show, however, that the elastic moduli are sensitive to structural changes that occur during and subsequent to the application of load cycles.

The factors controlling the frequency dependence of the elastic moduli also affect their temperature dependence at a given frequency. Reference GoldGold (1958) and Reference RamseierRamseier (unpublished) found a marked temperature dependence for Young's modulus between 0°C and –40°C for rate and duration of stress application equivalent to a frequency of stressing of less than 1Hz. This dependence is not observed for single-crystal ice stressed parallel or perpendicular to the basal plane. The behaviour is consistent with the hypothesis that deformation in the grain-boundary region contributes significantly to the strain for frequency of stressing less than about 10² Hz.

Observations on polycrystalline ice in the frequency range from 10² to 10³ Hz, show a maximum energy loss at 0°C, decreasing to a minimum between about –10°C to –30°C, and increasing to a second maximum at a lower temperature. The actual positions of the minimum and maximum depend on both frequency and purity (Reference Yamaji and KuroiwaYamaji and Kuroiwa, 1956; Reference KuroiwaKuroiwa, 1964). Kuroiwa gives evidence that the first maximum in the internal friction below about –30°C is associated with a simple energy-loss process having a single relaxation time. Good correspondence was found between the temperature dependence of the mechanical relaxation and that for the dielectric relaxation for the same frequency of excitation (i.e. about 300 Hz) indicating that this maximum in the internal energy loss is associated with the movement of proton defects in the ice crystal lattice. Reference VanDevender and ItagakiVanDevender and Itagaki (1973) argue from observations on the mechanical energy loss for single-crystal ice that the maximum also involves the movement of dislocations.

There is evidence of additional internal energy loss maxima at lower temperatures, which should correspond to relaxations in the elastic moduli at frequencies of stressing greater than 10³ Hz at –10°C. The information presented in Figure 2 shows that these changes for Young's modulus amount to probably less than 10% of its value at 10² Hz.

The dependence of the elastic moduli of polycrystalline ice on frequency of cyclic stressing, temperature, density, ice type, grain size, quality of the ice, and history of stressing, is still very imperfectly understood. From an engineering point of view, it would be reasonable to assume that for frequency of cyclic stressing greater than about 10³ Hz, the moduli could be considered to be independent of frequency and probably independent of temperature. For the frequency range below 10² Hz, however, both the frequency and temperature dependence may often have to be taken into consideration. Measurement of the temperature and frequency dependence of the moduli provides a potentially powerful technique for investigating the behaviour and influence of defects in the ice crystal structure, including grain boundaries, and for obtaining a measure of the quality of ice.

Engineering problems for which the frequency of stressing is less than 10 Hz are often associated with one cycle of load and an indefinite period until the next (e.g. the bearing-capacity problem for moving loads). It may be more appropriate for these cases to consider the dependence of the elastic moduli on strain-rate rather than frequency. Reference Gold, Traetteberg and WeaverGold and Traetteberg ([1975]) found from experiments for which the load was applied at essentially a constant rate to a maximum stress of about 0.5 MN m^-2, and immediately removed at the same rate, that the relaxation in Young's modulus for granular and columnar-grained (type S2) ice stressed perpendicular to the long direction of the grains, occurred over the same range in strain-rate as the ductile to brittle transition. They also found that the relaxation processes involved were not simple, and that the relaxation time increased with time of loading raised to the power of 2/3. It would appear that the strain-rate or frequency dependence of the elastic moduli of polycrystalline ice below 10² Hz is intimately related to the processes responsible for its yield and transition from a ductile to a brittle mode of behaviour.

Observations of Poisson's ratio for polycrystalline ice also provide evidence for deformation in the grain-boundary region contributing to the strain for frequency of stressing less than 10 Hz. Sonic measurements give a value of about 0.30 for the ratio. Measurements at low rate of stressing (load cycle completed in 5 s to 10 s) showed that for this condition Poisson's ratio does not remain constant but increases with strain (Reference GoldGold, 1958, Reference Gold1959). The ratio is probably dependent on the strain, strain-rate, grain structure, and stress geometry at low frequencies of stressing.

Failure

The bearing-capacity and ice-pressure problems require information on the strength of ice for given conditions of both stress and material. For the stress levels that are of concern, failure is preceded by or due to the formation of cracks either at the surface or in the interior. Considerable information has been obtained in recent years on the characteristics of crack formation in ice and the role that it plays in failure (Reference Carter and MichelCarter and Michel, 1971 ; Reference Hawkes and MellorHawkes and Mellor, 1972; Reference GoldGold, 1972[a], Reference Gold[b]). Initiation and propagation of cracks are found to depend on stress, strain, strain-rate, temperature, grain structure, grain size, shape of the body, and how the load is applied, and so it is not surprising that there is a large variation in reported values of the strength and a large measure of empiricism associated with strength or load calculations.

Tensile Strength

Fracture is usually the mode of tensile failure for polycrystalline ice at the stress levels of engineering interest. For this condition, failure is essentially brittle. The brittle tensile strength is almost independent of temperature for strain-rates greater than about 10^-5 s^-1. For uniaxial stress applied to granular ice, average grain size about 1 mm, temperature –10°C, its value is about 2 MN m^-2 and for columnar-grained ice with the stress perpendicular to the long direction of the grains, about 1 MN m^-2 (Reference Carter and MichelCarter and Michel, 1971). Reference Hawkes and MellorHawkes and Mellor (1972) observed a small strain-rate dependence for granular ice at –7°C with a maximum in the strength at a strain-rate of about 10^-3 s^-1.

Reference GoldGold (1972[a], Reference Gold[b]) found that the formation of transcrystalline cracks during ductile deformation of ice is consistent with models of crack initiation based on dislocation pile-ups, such as that by Reference Smith and BarnbySmith and Barnby (1967). A criterion for initiation derived by them is:

(3)

where τ_e is the effective shear stress acting on a pile-up of length l, G is shear modulus, v is Poisson's ratio, and γ is the energy associated with the formation of new surface.

If it is assumed that l is one-half the average grain diameter and equal to 1 mm;

then τ_e = 0.65 MN m^-2. This would correspond to an applied uniaxial stress of about 1.3 MN m^-2 which is in reasonable agreement with the minimum stress at which cracks are observed to form in ice (about 0.6 MN m^-2 for columnar-grained ice, average grain diameter about 3 mm (Reference GoldGold, 1972[a])).

Because of the characteristics of crack initiation and propagation, ice under stress can be in a “metastable” state; that is, in a stress state for which fracture would occur if a large enough crack or flaw were present. The crack that is initiated by a stress concentration such as a dislocation pile-up will have a size that is determined by the local strain energy. It will be a stable, non-propagating crack if it is not sufficiently big to be propagated by the applied stress.

For columnar-grained ice, it is reasonable to assume for a uniaxial tensile stress that the crack of critical size for propagation under the applied stress would be long and narrow and subject to the condition of plane stress. The criterion for its propagation is given by:

(4)

where σ_f is the tensile stress, E is Young's modulus, 2c is the crack width, and γ_p is the surface energy (0.1 J m^-2) plus the work associated with viscous or plastic deformation during propagation (see Reference Tetelman and McEvilyTetelman and McEvily, 1967).

There is little information available on γ_p for ice. Observations by Reference GoldGold (1963) on crack formation by thermal shock indicate that it is not too different from γ, the surface energy. This is consistent with the observation that formation of cracks in ice during creep or constant strain-rate tests has the characteristics of a brittle event.

If it is assumed for columnar-grained ice that σ_f = 1 MN m^-2, E = 7 × 10³ MN m^-2, γ_p = 0.9 J m^-2, then Equation (4) gives 2c = 0.9 mm and for σa_f = 0.6 MN m^-2, 2c = 2.5 mm.

These values, which are of the same order as the grain size, are the smallest crack widths that would result in fracture at the respective uniaxial tensile stresses. Trans-crystalline cracks of this size are observed to form in columnar-grained, S2 ice during uniaxial compressive creep for the same stresses (Reference GoldGold, 1972[a]). The strain to the formation of the first cracks of this size is about 10^-3 and 5 × 10^-3 respectively.

For granular ice it would be reasonable to assume that the critical crack would be “pennyshaped”. The criterion for its propagation is given by:

(5)

assuming σ_f = 2 MN m^-2, E = 6 × 10³ MN m^-2, v = 0.35 and γ_p = 0.1 J m^-2 gives 2c = 0.54 mm, and for σ_f = 0.6 MN m^-2, 2c = 6 mm. It would be expected that the size of a dislocation crack would be normally limited to about the grain size. Observations by Reference Hawkes and MellorHawkes and Mellor (1972) on the tensile strength of granular ice of average grain size 0.7 mm are consistent with the above calculations. They found that for uniaxial tensile stress less than about 1 MN m^-2, the maximum strength was associated with yield rather than fracture. They also found that the strain to fracture was less than 10^-3 and decreased with increasing strain-rate, tending to about 3 × 10^-4 for strain-rate greater than 10^-3 s^-1. The foregoing calculations indicate that initiation and propagation of cracks in ice for the values of tensile stress of interest for engineering problems is consistent with current theoretical models and with an energy of formation of new surface during crack propagation that is about equal to the surface energy of ice.

A question of some interest is the time and temperature dependence of fracture. One school of thought that has received appreciable attention is that fracture is a thermally activated process (Reference Zhurkov and SanfirovaZhurkov and Sanfirova, 1960). Such processes must play a role, for example, in the movement of dislocations into and out of pile-ups, and in their ultimate coalition to initiate a crack. But the influence of time and temperature on the fracture strength is probably not direct, but rather through their effect on the deformation processes that are responsible for the build-up of crack-induced stress concentrations. For stress greater than 1 MN m^-2 for columnar-grained, S2 ice, for example, strain to the formation of the first large crack is relatively constant and about equal to 2 × 10^-4 (Reference GoldGold, 1972[a]). Reference Hawkes and MellorHawkes and Mellor (1972) found the strain to tensile fracture for granular ice subject to uniaxial tension, decreased from about 6 × 10^-4 at a strain-rate of 10^-5 s^-1, to about 3 × 10^-4 at a strain-rate of 10^-3 s^-1. Time to failure for both these situations would depend primarily on the strain-rate. An analysis of the temperature and stress dependence of the time to failure would, therefore, reflect the dependence of the strain-rate on these factors. Activation energies determined by Reference GoldGold (1972[a]) from observations on the time to formation of first cracks did, in fact, have values that were about equal to those obtained from the temperature dependence of the strain-rate for a given stress.

Few engineering problems involve the uniaxial tensile strength of ice. Problems for which the tensile strength is important are usually associated with the vertical deflection of ice covers, and so the property of interest is the flexural strength for the biaxial stress condition associated with the bending of plates. Measurements of flexural strength of ice, however, are normally made on beams and at a rate at which the visco-elastic behaviour is significant. In calculating this strength, little attention has been given to the possible effects of crack-initiating stress concentrations at the surface of beams or to the strain-rate dependence of the elastic modulus and of the strength itself.

It is of some interest to note that tensile strengths determined from centre-loaded simply-supported circular ice plates, for which the two in-plane principal stresses in the region of maximum stress are equal, can be appreciably greater than those obtained from uniaxial tests. The author has measured values between 3.55 and 4.55 MN m^-2 (8 tests) for ice plates 50 cm in diameter and about 1.25 cm thick, loaded to failure within 10 s.

Compressive strength

Information on the strain-rate dependence of the maximum strength of various types of polycrystalline ice subject to a uniaxial compressive stress is presented in Figure 3. For the columnar-grained ice, the stress was applied perpendicular to the long axis of the columns. It can be seen that the compressive strength has a maximum at a strain-rate of between 10^-4 to 10^-3 s^-1. There is still some uncertainty concerning the decrease at higher strain-rates as this is a particularly difficult region for measurements and measured values may be significantly affected by the conditions between the load platens and the ends of specimens. For lower rates of strain up to at least 10^-5 s^-1, the dependence of the yield stress on the strain-rate on a constant strain-rate test is in good agreement with the dependence of the secondary creep rate on the stress in a constant load test (see Fig. 3) .

Fig 3.

Strain rate dependence of the yield and brittle strength (● ○ ■□ + ×); stress dependence of secondary creep rate (▲, △).

It is of interest that the compressive strengths of columnar type S2 ice, granular ice and ice formed from frazil are about the same for a given rate of strain. Each of these ice types will have a resolved shear stress on the basal planes of most of the grains, and mobile dislocations on this plane of easy slip will be able to participate readily in the initiation and development of new modes of deformation. Columnar type S I ice, for which the basal planes are perpendicular to the long direction of the grains and, therefore, parallel to the applied stress (i.e. there is no resolved shear stress on the basal planes), has a significantly higher strength for strain-rates less than 10^-3 s^-1. Figure 4 illustrates clearly the difference that exists between the deformation behaviour of the various ice types in the ductile range. Type S2 ice has a relatively sharp yield within the first 0.5% strain. This appears to be due to a relatively rapid build-up of internal stress concentrations under the constraints associated with this particular columnar-grained structure (Reference GoldGold, 1972[a]). Granular ice (type TI) and frazil ice (type S4) are more ductile with yields at about 1 % strain. They also appear to be tougher, which is probably due to their granular structure, although grain size may also have an effect. Columnar-grained ice, type SI, is much stronger, less ductile in its behaviour, with yield at about 0.5%.

Fig 4.

Strain dependence of the stress for ice types from the St Lawrence River. Strain-rate, 1.67 × 10^-5 s^-1; temperature, –9.5°C (Reference Gold and KrauszGold and Krausz, 1971).

The formation of internal cracks is an important feature for compressive yield and fracture of polycrystalline ice for uniaxial stress greater than about 1.0 M m^-2. For maximum stresses below that value, crack formation is essentially random, indicating little interaction between cracks, and confined primarily to the primary creep or pre-yield stages. Above that value, crack formation is initially random, but usually becomes non-uniform prior to yield or failure. The pattern of the cracking activity depends on the condition of the platen surfaces and the ends of the specimens. Under certain conditions, specimens fail by axial cleavage. If the ends of the specimens are sufficiently constrained, shear failure will occur, with crack formation concentrated in zones parallel to the planes of maximum shear (Reference Hawkes and MellorHawkes and Mellor, 1972; Reference GoldGold, 1972[a]).

If a polycrystalline ice specimen is subject to biaxial rather than uniaxial compressive stress, crack formation and the time-dependent strain are suppressed in the direction of the constraining force, and the strength is modified accordingly. Figure 5 presents observations on the strength of columnar-grained type S2 ice rigidly confined in a direction perpendicular to that of the applied load (Reference FrederkingFrederking, [1972]). They clearly show the anisotropy of the strength of this ice type. The effect of constraint is minimal for granular ice which is still able to flow relatively freely in the third orthogonal direction. For the columnar-grained ice, the strength is approximately doubled if the constraining force is perpendicular to the long direction of the columns; there is little effect if it is in the long direction of the columns.

Fig 5.

Strain-rate dependence of yield and failure stress for columnar-grained type S2 ice subject to compressive stress. For A-type specimens, applied stress and confining stress perpendicular to long direction of grains; for B-type specimens, applied stress perpendicular to long direction, confining stress parallel to it. T = –100° (Reference FrederkingFrederking, [1972]).

The characteristics of crack formation in ice are significant for many engineering problems involving a structure subject to a load from an ice cover. It is commonly assumed that the maximum force that can be exerted by the ice is that associated with its crushing strength as determined from uniaxial compression tests. This may be true, for example, for failure at a cylindrical structure when cracks radiating from the structure divide the ice cover into wedge-shaped pieces which then fail by a combination of shear and crushing (Reference Croasdale, Reed and SaterCroasdale, 1974). Under some conditions, however, the maximum force may be determined by the formation of one or more cleavage cracks parallel to the surface of the ice subsequent to the formation of cracks normal to the surface as observed in laboratory experiments (Reference Zabilansky, Zabilansky, Nevel and HaynesZabilansky and others, 1975), Reference Hirayama, Hirayama, Schwarz and WuHirayama and others (1974) present evidence that crack formation in the ice near a structure is constrained due to friction or adhesion between the structure and the ice. A “plastic” zone is induced in the ice adjacent to the structure and because of the lateral restraint of the surrounding ice the flow in it tends to be normal to the surface of the ice cover. They argue that the cleavage cracks are due to the tensile stresses induced by this plastic flow. This type of failure can be observed in the laboratory if the specimen is constrained in a direction perpendicular to the applied load or if there is little or no friction between the specimen and the load pattern. Information is now becoming available that, it is to be hoped, will allow the appropriate relationships to be established between the strength and deformation properties of ice measured at a laboratory scale and those that determine its behaviour when moving against a structure (Reference Wu, Wu, Chang and SchwarzWu and others, 1974; Reference Croasdale, Reed and SaterCroasdale, 1974). As shown by Wu and others, the stress immediately in front of the structure depends on the width of the structure and the thickness of the ice in addition to the properties of the ice under given conditions.

Reference Carter and MichelCarter and Michel (1971) found the tensile strength of polycrystalline ice to be little influenced by temperature between 0°C and –40°C for the brittle range of behaviour. This indicates that the quantity Eσ _ρ/c in Equations (4) and (5) is essentially independent of temperature. They did find, however, that the strain-rate associated with the transition is temperature dependent, decreasing with decreasing temperature and being between 10_-6 and 10_-7 s_-1 at –10°C.

The situation is different for the compressive strength of polycrystalline ice. Failure in the ductile range in this case is associated with the deterioration of the structure through the formation of essentially non- or slowly-propagating cracks. The formation of a crack indicates that a critical condition was attained, a condition that depends on structure, relative crystallographic orientation of adjacent grains, and the relative contribution of the various modes of deformation. The latter are quite dependent on temperature. Observations by Reference GoldGold (1972[b]) have shown that the number of cracks that form in a given strain for the same constant compressive stress, decreases with decreasing temperature. For tests being carried out at constant strain-rate, this would have the effect of causing an apparent increase In strength with decrease in temperature (Reference Carter and MichelCarter and Michel, 1971 ; Reference ButkovichButkovich, 1954).

Compressive strength cannot be simply related to temperature. The temperature dependence also depends on the type of ice, strain-rate, and grain size. An appreciation of the factors controlling the strength of ice is only now beginning to develop. Much work on this problem has still to be done.

Ductile to brittle transition

A question that is of some interest to the engineer is the transition from ductile to brittle behaviour for polycrystalline ice. In tension, this transition occurs relatively abruptly, i.e. when conditions are such that a crack of sufficient size to propagate as a Griffith crack is initiated before yield occurs. This condition depends on the strain-rate, stress, temperature, ice type and grain size. The tensile stress associated with brittle behaviour for columnar-grained S2 ice, average grain size 3 mm, subject to a force perpendicular to the long direction of the grains, is about 1.0 MN m^-2 and for granular ice of average grain size about I mm, 2.0 MN m^-2 (see Reference Carter and MichelCarter and Michel, 1971; Reference Hawkes and MellorHawkes and Mellor, 1972). Hawkes and Mellor found that although the brittle tensile strength is relatively independent of strain-rate, it does appear to have a maximum at about 10^-3 s^-1. The maximum indicates that even in the brittle range viscous flow must still be associated with fracture. Totally elastic response appears to occur only for strain-rates greater than about 10^-1 to 1 s^-1. This would correspond to a rate of cyclic stressing greater than about 100 Hz.

It does appear that the apparent maximum in the brittle tensile strength occurs in the same range of strain-rate as that which would be associated with the change in Young×s modulus over the frequency range of 10^-3 to 100 Hz shown in Figure 2. The possible relationship between these two features has not yet been investigated theoretically or experimentally. This behaviour, however, is consistent with the stress dependence of the dislocation velocity. At a strain-rate of 10^-5 s^-1, a dislocation would move in granular ice about 10^-2 cm before failure, assuming the validity of Equation (2). This is appreciably less than the average grain size of fine-grained ice, i.e. about 0.1 cm. Observations of the Young's modulus of granular ice indicate that it is essentially elastic to failure in tension for strain-rate equal to or greater than this value for temperature lower than about –5°C (Reference Carter and MichelCarter and Michel, 1971; Reference Hawkes and MellorHawkes and Mellor, 1972). The foregoing indicates that the ductile to brittle transition in tension occurs when the strain-rate is such as not to allow sufficient time for dislocations to contribute significantly to the strain prior to the initiation of fracture. The relative constancy of the brittle tensile strength suggests that fracture begins at sites whose effectiveness is relatively independent of the strain-rate.

The ductile to brittle transition is much more gradual for compressive stress. The initiation of a crack during the initial stage of deformation does not affect the strength significantly, but it does allow the ice in the vicinity of it to conform more easily to the imposed deformation, i.e. a form of work softening. This cracking activity must certainly be a significant factor with respect to yield for ductile behaviour and stress greater than about 1 MN m^-2 (Reference GoldGold, 1972[a], Reference Gold[b]). One of the interesting features of this activity is that it appears to be composed of two independent populations; one with a probability of crack initiation directly proportional to the strain, and the second with a probability of initiation independent of the strain. The former appears to be associated with trans-crystalline cracks and is consistent with crack initiation by dislocation pile-ups; the second with grain-boundary cracks.

The two populations appear to have different dependencies on strain-rate. There is evidence that grain-boundary cracks dominate in brittle fracture (Reference GoldGold, 1972[a]; Reference Wu, Wu, Chang and SchwarzWu and others, 1974). As the majority of cracks are transcrystalline for compressive creep at stress below 2 MN m^-2, there must be a transition in the characteristics of the cracking activity. This transition has yet to be explored experimentally.

The ductile to brittle transition in polycrystalline ice subject to uniaxial compression appears to be associated with this change in the characteristics of the cracking activity and their dependence on stress, strain-rate, temperature, type of ice and conditions between the load platens and the ends of the specimen. If the strain-rate is sufficiently large, enough cracks form in the loading phase, or a major cleavage crack is initiated at a site of stress concentration, such as at the specimen end, so that failure occurs before yield. For this condition the stress- strain curve is essentially linear and failure can be explosive. But even at the rate of strain of 10^-3 s^-1 at –10°C, time to failure is about 1 s, so there must be significant visco-e1astic deformation. With further increase in strain-rate the deformation becomes more elastic and the strength more sensitive to pre-existing stress-raising imperfections. The decrease in strength for strain-rate greater than about 10^-3 s^-1 is very probably due to the decrease in the work of fracture because of the increased elastic behaviour of ice.

To date, most attention has been directed to the strain-rate dependence of the strength of the various types of ice. Much research is still to be done. The deformation behaviour of ice in the range of strain-rate associated with the ductile to brittle transition and its dependence on all possible factors, is still not fully understood. For example, little is known on how the cracking activity responsible for failure depends on type of ice, temperature, conditions of loading, and grain size, or how the failure process depends on the size of specimen. Little is known also about the ductile to brittle transition for biaxial and triaxial stress conditions, which are the normal conditions for field situations. The failure of ice has still to be investigated using recently developed techniques of fracture mechanics.

Concluding remarks

This review has given attention primarily to the deformation and strength behaviour of ice, behaviour that is of particular relevance to two important ice engineering problems: ice pressure on structures and the bearing capacity of ice covers. Because of limitations of space no mention has been made of other problems of concern to engineers, such as the adhesion and build-up of ice on structures and road surfaces, frictional effects on ice surfaces, and the determination of the thickness of ice bodies including ice covers. The latter need has resulted in the recent development of techniques based on the response of ice to energy propagated in various ranges of the electromagnetic spectrum. This work is producing detailed knowledge concerning the electrical properties of ice as well as opening up new possibilities for remote sensing of the characteristics of ice covers (Reference Page and RamseierPage and Ramseier, 1975). The frictional behaviour of the ice surface is still not properly understood, even though it is a major factor in the sports of skating and curling and for the safety of roads and airstrips in winter. The science and a rt of obtaining solutions to engineering problems involving ice have still not attained an acceptable level of maturity.